A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

@article{Williams2015ADA,
  title={A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition},
  author={Matthew O. Williams and Ioannis G. Kevrekidis and Clarence W. Rowley},
  journal={Journal of Nonlinear Science},
  year={2015},
  volume={25},
  pages={1307-1346}
}
The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar… 
Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition
TLDR
A brief summary of the Koopman operator theorem for nonlinear dynamics modeling is provided and several data-driven implementations using dynamical mode decomposition (DMD) for autonomous and controlled canonical problems are analyzed.
Dynamic Mode Decomposition with Reproducing Kernels for Koopman Spectral Analysis
TLDR
A modal decomposition algorithm to perform the analysis of the Koopman operator in a reproducing kernel Hilbert space using finite-length data sequences generated from a nonlinear system is proposed.
Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems
TLDR
This work considers the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems, and demonstrates the impact of observable selection, including kernel methods, and construction of the Koop man operator on several canonical nonlinear PDEs.
Scalable Extended Dynamic Mode Decomposition Using Random Kernel Approximation
TLDR
An approach to EDMD in which the features used provide random approximations to a particular kernel function, and two specific methods for generating features: random Fourier features, and the Nystrom method are discussed.
Efficient Identification of Linear Evolutions in Nonlinear Vector Fields: Koopman Invariant Subspaces
TLDR
The Symmetric Subspace Decomposition strategy is introduced to identify linear evolutions using efficient linear algebraic methods and an alternative characterization based on the identification of the largest Koopman invariant subspace in the span of the dictionary is proposed.
A kernel-based method for data-driven koopman spectral analysis
TLDR
A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented, using a set of scalar observables that are defined implicitly by the feature map associated with a user-defined kernel function.
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator
TLDR
This work establishes the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator, and shows that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables.
Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition
  • Bowen Huang, U. Vaidya
  • Computer Science, Mathematics
    2018 Annual American Control Conference (ACC)
  • 2018
TLDR
A new algorithm for the finite dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time series data is provided, described as naturally structured DMD since it retains the inherent properties of these operators.
Delay-Coordinate Maps and the Spectra of Koopman Operators
The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 71 REFERENCES
Koopman-mode decomposition of the cylinder wake
  • S. Bagheri
  • Engineering
    Journal of Fluid Mechanics
  • 2013
Abstract The Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow
Spectral analysis of nonlinear flows
We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an
Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses
TLDR
It is shown that expansion in DMD modes is unique under certain conditions, and an “optimized” DMD is introduced that computes an arbitrary number of dynamical modes from a data set and is superior at calculating physically relevant frequencies, and is less numerically sensitive.
Optimal mode decomposition for unsteady flows
TLDR
The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized but rather fixed to be the proper orthogonal decompose (POD) modes.
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions
In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of
Dynamic mode decomposition of numerical and experimental data
  • P. Schmid
  • Physics, Engineering
    Journal of Fluid Mechanics
  • 2010
The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information
Spectral signature of the pitchfork bifurcation: Liouville equation approach.
  • Gaspard, Nicolis, Provata, Tasaki
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1995
TLDR
It is shown that the Liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation, and the spectral decompositions constructed here for theLiouville equation are obtained as the noiseless limit of the well known spectral decomposition of the Fokker-Planck equation of the associated stochastic process.
Applied Koopmanism.
TLDR
The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice, and the paper highlights its strengths in applied and numerical contexts.
On dynamic mode decomposition: Theory and applications
TLDR
A theoretical framework in which dynamic mode decomposition is defined as the eigendecomposition of an approximating linear operator, which generalizes DMD to a larger class of datasets, including nonsequential time series, and shows that under certain conditions, DMD is equivalent to LIM.
...
1
2
3
4
5
...